Approximate/Fuzzy Search for strings keys

I have read the literature on fuzzy/approximate search for strings
keys in large databases, but I could not find a clear answer on what
is the best that has been achieved thus far in terms of the cost of
search (regardless of cost of building any index).

I understand this depends on the application/domain/type of data, but,
to keep things simple, supoose the target data is some string (names,
addresses, or any sequence of characters), and assume any distance
function. I am simply asking this: what is the best that has been
achieved? Is there a logarithmic solution (i.e., that can retrieve a
set of "similar" objects in some logarithmic order), or are they all,
in the worst case, linear?